Time domain signal of STF and LTF has period 0.8us and 3.2us respectively. With this knowledge, frequency error can be computed through shifted cross-correlation operation.
STF signal assist in computing coarse estimate of frequency error and a fine estimate can be obtained from LTF.
Frequency Error Estimate Range
Range of frequency errors that can be computed from STF will be [-1/(2*0.8u) to 1/(2*0.8u)] or [-625khz to +625khz]
Range of frequency errors that can be computed from LTF will be [-1/(2*3.2u) to 1/(2*3.2u)] or [-156.25khz to +156.25khz]
Above ranges can be obtained by equating the angle of \(e^{j2\pi f T_p}\) to \(+/- \pi\). Where \(T_p\) is the period of STF or LTF.
Frequency Error Estimate Resolution
Resolution of frequency error estimate has significance in real time systems. Resolution of fixed point implemenation determine the resolution of the frequency error estimation algorithm.
A \('n' \) bit precision fixed point implementation will have an ampitude resolution of \(2^{-n}\). To know the resolution of frequency error estimate, it is required to know the smallest possible phase represented in the implemented fixed point.
Let us say \(\cos\left(x\right)\) is well represented by the fixed point. If \(\Delta = (2 \pi f T_p)\) be the possible phase resolution by the fixed point, then \(\cos\left(x +\Delta\right)~ - ~\cos\left(x\right) = -2^{-n}\) gives rise to \(\Delta \sin\left(x\right) = 2^{-n}\).
Similarly, Let us say \(\sin\left(x\right)\) is well represented by the fixed point, then \(\sin\left(x +\Delta\right)~ - ~\sin\left(x\right) = 2^{-n}\) gives rise to \(\Delta \cos\left(x\right) = 2^{-n}\).
From the above \( \Delta = +/- \sqrt{2} . 2^{-n}\) or \( f = +/- \sqrt{2} . 2^{-n} . \frac{1}{2 \pi T_p}\)
